Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Add the chicken pieces to the skillet and cook until lightly browned on all sides, about 5-6 minutes.
Add the sliced mushrooms, diced carrots, and frozen peas to the skillet. Cook for 5 minutes until vegetables begin to soften.
Sprinkle the flour over the chicken and vegetables, stirring well to coat everything evenly. Cook for 2 minutes to remove the raw flour taste.
Gradually pour in the chicken broth while stirring continuously to avoid lumps.
Add the heavy cream, dried thyme, dried parsley, salt, and black pepper. Stir to combine and bring the mixture to a gentle simmer. Cook for 5 minutes until the sauce thickens.
Remove the skillet from heat and stir in the cooked white rice.
Transfer the mixture to a greased 9×13 inch casserole dish. Sprinkle the shredded cheddar cheese and grated Parmesan evenly over the top.
Bake in the preheated oven for 20 minutes or until the cheese is melted and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.